194 

 Assume now, 



A 



APPENDIX. 



A = c C d<f> = 1-15921 () 



o 

 a 



B = c C d<}, (cos. <f>) 



o 

 a 



A' = cC d<j> (cos. 3 <) = 1-15921 ( 3 s ' a + 



= 1-15921 (sin. a) 



3 sin. a sin. 3 a \ 



12 



B' = c d<f> (cos. 2 <) = 1-15921 + 



and we shall have, 



With the COMMON WHEEL, 



f expended by the Engine 

 Horses' Powers 



L effective on the Vessel 



1000 



V(AV 2 -2BV0+ BVn 



. .(m); 



With the VERTICALLY ACTING WHEEL, 



("expended by the Engine 

 Horses' Powers 



L effective on the Vessel 



(A'V 2 -2B'Vt>+ 



} 



(*). 



These last expressions afford great facility in the solution of any questions concerning the 

 working of paddle wheels, as they give the most simple relations amongst the four leading 

 quantities, viz., the proportionate immersion, the power, and the two velocities, and enable us 

 to determine any one of these when the other three are known. 



For convenience of calculation the values of the coefficients A, B, A', B', which depend on 

 the immersion of the centre of pressure, are exhibited in the following table. They are given 

 for every hundredth of immersion as far as eight-tenths of the radius of the wheel, which will 

 take in every possible case in practice ; and the smaller figures inserted at the right hand of each 

 column are the respective differences of the two values between which they stand, and serve to 

 expedite the interpolation of any intermediate value falling between those in the table, for 

 which simple proportion will be quite sufficient. 



